Algebraic Manipulation

Algebraic simplification is useful in its own right. Within SymPy, though, it is mostly used indirectly as a tool in other areas of the library. In fact, many mathematical problems in symbolic computing are first expressed using entities from the symbolic core before they are preprocessed and transformed into a problem in polynomial algebra, where generic and efficient algorithms are used to solve the problem. The solution to the original problem is then subsequently recovered from the results.

Simplification#

The generic way to simplify an expression is by calling the simplify() function. It must be emphasized that simplification is not a rigorously defined mathematical operation. The simplify() function applies several simplification routines along with heuristics to make the output expression simple.

In some cases, applying simplify() may actually result in a more complicated expression.

Due to this issue, we use specialized simplification methods. One example that we will discuss below is trigsimp().

Simplifying trigonometric expressions#

The trigsimp() method is a specialized method used to simplify trigonometric expressions. Let’s see an example of this below:

Expand#

The expand() method expands the mathematical expression. Let’s see an example of this below:

See help(expand) to explore the various types of expansions that expand() can perform.

Factor#

The opposite of a product expansion is, of course, factoring. The factor() method factors a polynomial into irreducibles and returns them. Let’s look at an example of this below:

Collect#

The collect() method collects the polynomial coefficients of the symbol specified as the second argument of the method. The first argument is the expression itself. Let’s look at an example of this below:

Fractions#

Decomposition#

apart() computes the partial fraction decomposition of a rational function:

Combination#

The together() method combines rational functions into a single fraction and factors it.

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